I'm having similar problems, but probably not for the same reasons.

Function 'Sleep' could not be resolved	main.cpp	/sfmltest	line 22	Semantic Error
Invalid overload of 'endl'	main.cpp	/sfmltest	line 21	Semantic Error
Method 'GetElapsedTime' could not be resolved	main.cpp	/sfmltest	line 19	Semantic Error
Method 'GetElapsedTime' could not be resolved	main.cpp	/sfmltest	line 21	Semantic Error
Type 'sf::Clock' could not be resolved	main.cpp	/sfmltest	line 18	Semantic Error

#include <SFML/System.hpp>
#include <iostream>
//#include <iomanip>

using namespace std;

int main()
{
    sf::Clock Clock;
    while (Clock.GetElapsedTime() < 5.f)
    {
        cout << Clock.GetElapsedTime() << endl;
        sf::Sleep(0.5f);
    }

    return 0;
}

I'm attempting to configure SFML to work with Eclipse.

What's funny is that it does compile and link and I can run the executable from the Debug folder and it works exactly as it should, but I still get these errors, and in the editor it shows errors and red underlines in the code.

I've been using Eclipse CDT for the last 2 years at school, but this is the first time I've tried to use an external library.

These are not build errors (from the compiler) but "semantic errors". The internal code analyzer (CODAN) creates them. CODAN depends on the CDT index that represents the structure of your source code with all declarations, definitions etc. (the index is used e.g. to get code completion, jump to to the declaration/definition with the key F3 etc.)
In your case it looks like the index is not complete. You should add the include paths for your external library to Project Properties->C/C++ general->Paths and Symbols and then rebuild the index (from the project's context menu).