Here is the trace:
<Error>
   <Message>An error has occurred.</Message>
   <ExceptionMessage>
      Type 'ProjectName.Web.Api.Controllers.ContinentsController' does not have a default constructor
   </ExceptionMessage>
   <ExceptionType>System.ArgumentException</ExceptionType>
   <StackTrace>
      at System.Linq.Expressions.Expression.New(Type type)
      at System.Web.Http.Internal.TypeActivator.Create[TBase](Type instanceType)
      at System.Web.Http.Dispatcher.DefaultHttpControllerActivator.GetInstanceOrActivator(HttpRequestMessage request, Type controllerType, Func`1& activator)
      at System.Web.Http.Dispatcher.DefaultHttpControllerActivator.Create(HttpRequestMessage request, HttpControllerDescriptor controllerDescriptor, Type controllerType)
   </StackTrace>
</Error>
I find this weird as public class UsersController : ApiController { ... } is working just fine. I have compared the 2 controllers, all the settings and structures are similar.
I am using Ninject and i have my system setup similar to Jamie Kurtz Asp.Net Mvc 4 and the Web Api: Building a REST Service from Start to Finish.
From the stack trace, is anyone able to spot the problem and how to solve it? Thanks!
As Requested.
ContinentsController
[LoggingNHibernateSession]
public class ContinentsController : ApiController
{
    private readonly ISession _session;
    private readonly IContinentMapper _continentMapper;
    private readonly IHttpContinentFetcher _httpContinentFetcher;
    private readonly IDateTime _dateTime;
    public ContinentsController(ISession session, IContinentMapper continentMapper, IHttpContinentFetcher continentFetcher, IDateTime dateTime)
    {
        _session = session;
        _continentMapper = continentMapper;
        _httpContinentFetcher = continentFetcher;
        _dateTime = dateTime;
    }
    public IEnumerable<Continent> Get()
    {
        var continents = _session
            .Query<Data.Model.Continent>()
            .Select(_continentMapper.CreateContinent)
            .ToList();
        return continents;
    }
    public Continent Get(long id)
    {
        var modelContinent = _httpContinentFetcher.GetContinent(id);
        var continent = _continentMapper.CreateContinent(modelContinent);
            return continent;
        }
  }
UsersController: Works just fine.
    public class UsersController : ApiController
    {
        private readonly ISession _session;
        private readonly IUserManager _userManager;
        private readonly IUserMapper _userMapper;
        private readonly IHttpUserFetcher _userFetcher;
        public UsersController(
            IUserManager userManager,
            IUserMapper userMapper,
            IHttpUserFetcher userFetcher,
            ISession session)
        {
            _userManager = userManager;
            _userMapper = userMapper;
            _userFetcher = userFetcher;
            _session = session;
        }
        [Queryable]
        public IQueryable<Data.Model.User> Get()
        {
            return _session.Query<Data.Model.User>();
        }
        [LoggingNHibernateSession]
        public User Get(Guid id)
        {
            var user = _userFetcher.GetUser(id);
            return _userMapper.CreateUser(user);
        }
    }
I am using NinjectWebCommon.cs and in it, i have this and a few other default methods.:
        private static void RegisterServices(IKernel kernel)
        {
            var containerConfigurator = new NinjectConfigurator();
            containerConfigurator.Configure(kernel);
            GlobalConfiguration.Configuration.MessageHandlers.Add(kernel.Get<BasicAuthenticationMessageHandler>());
        }
Then i have NinjectConfigurator.cs:
public class NinjectConfigurator
{
    ......
    private void AddBindings(IKernel container)
        {
            .....
            container.Bind<IDateTime>().To<DateTimeAdapter>();
            container.Bind<IDatabaseValueParser>().To<DatabaseValueParser>();
            //HttpFetchers
            container.Bind<IHttpUserFetcher>().To<HttpUserFetcher>();
            container.Bind<IHttpContinentFetcher>().To<HttpContinentFetcher>();                                
            //TypeMappers
            container.Bind<IUserManager>().To<UserManager>();
            container.Bind<IMembershipInfoProvider>().To<MembershipAdapter>();
            container.Bind<IUserMapper>().To<UserMapper>();
            container.Bind<IContinentMapper>().To<ContinentMapper>();    
            .........
        }
     .......
}
Both NinjectWebCommon.cs and NinjectConfigurator.cs are located in the App_Start folder. 
container.Bind<ISession>().ToMethod(CreateSession); is NHibernate. It is within NinjectConfigurator.cs inside private void ConfigureNHibernate(IKernel container) { ... }
You need to tell Ninject how to correctly resolve Web API dependencies.
You can use Felipe Oriani's answer, but if you like there is a NuGet package called WebApiContrib.IoC.Ninject which will do this for you.
In Visual Studio Go to: Tools > NuGet Package Manager > Manage NuGet Packages for Solution
Install the WebApiContrib.IoC.Ninject package
Edit: NinjectWebCommon.cs and update the CreateKernel() method to include: GlobalConfiguration.Configuration.DependencyResolver = new NinjectResolver(kernel);
private static IKernel CreateKernel()
{
    var kernel = new StandardKernel();
    try
    {
        kernel.Bind<Func<IKernel>>().ToMethod(ctx => () => new Bootstrapper().Kernel);
        kernel.Bind<IHttpModule>().To<HttpApplicationInitializationHttpModule>();
        RegisterServices(kernel);
        //Note: Add the line below:
        GlobalConfiguration.Configuration.DependencyResolver = new NinjectResolver(kernel);
        return kernel;
    }
    catch
    {
        kernel.Dispose();
        throw;
    }
}
This error is a known error when you do not set the dependency resolver in your application. The Controller Factory could not find a parameterless constructor to create the controller and execute the action method (or verb method?). So, you have to create a dependency resolver class and set it on the initialization of your web app. It will resolve dependencies of your controllers.
Using ninject, you could try something like this:
using Ninject; 
using Ninject.Syntax; 
using System; 
using System.Collections.Generic; 
using System.Diagnostics.Contracts; 
using System.Web.Http.Dependencies; 
namespace MyApplication.App_Start 
{     
    public class NinjectDependencyScope : IDependencyScope     
    {         
        private IResolutionRoot resolver;         
        internal NinjectDependencyScope(IResolutionRoot resolver)         
        {             
            Contract.Assert(resolver != null);             
            this.resolver = resolver;         
        }         
        public void Dispose()         
        {             
            IDisposable disposable = resolver as IDisposable;             
            if (disposable != null)                 
                disposable.Dispose();             
            resolver = null;         
        }         
        public object GetService(Type serviceType)         
        {             
            if (resolver == null)                 
                throw new ObjectDisposedException("this", "This scope has already been disposed");             
            return resolver.TryGet(serviceType);         
        }   
        public IEnumerable GetServices(Type serviceType)         
        {             
            if (resolver == null)                 
                throw new ObjectDisposedException("this", "This scope has already been disposed");             
            return resolver.GetAll(serviceType);         
        }     
    }     
    public class NinjectDependencyResolver : NinjectDependencyScope, IDependencyResolver     
    {         
        private IKernel kernel;         
        public NinjectDependencyResolver(IKernel kernel)             
            : base(kernel)         
        {             
            this.kernel = kernel;         
        }         
        public IDependencyScope BeginScope()         
        {             
            return new NinjectDependencyScope(kernel.BeginBlock());         
        }     
    } 
}
The NinjectDependencyResolver class takes a Ninject StandardKernel object as a constructor argument and this reference is used whenever a dependency scope is pipelined.
To make this all work, the NinjectDependencyResolver class is assigned to the application's global configuration:
private static IKernel CreateKernel()
{
    var kernel = new StandardKernel();
    kernel.Bind<Func<IKernel>>().ToMethod(ctx => () => new Bootstrapper().Kernel);
    kernel.Bind<IHttpModule>().To<HttpApplicationInitializationHttpModule>();
    // register all your dependencies on the kernel container
    RegisterServices(kernel);
    // register the dependency resolver passing the kernel container
    GlobalConfiguration.Configuration.DependencyResolver = new NinjectDependencyResolver(kernel);
    return kernel;
}
On your Global.asax.cs file in the end of Application_Start event, call this CreateKernel method.
For me, it was just a matter of adding the Ninject.Web.WebApi.Webhost NuGet package.
For my applications that use both MVC and WebApi2, I have the following packages for Ninject:
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