How do you get a ship?

Supposedly you can get your own spaceship, but how do I go about doing that? I am level 13, and have left the beginner area ages ago.

I have checked the hangar, at republic fleet, but it just said there was no ship for me there, and wouldn't let me enter.

Solution 1:

After completing the last class quest (about level 16) on your second planet you will receive your ship.

You will recive a space ship after the following quest:

Planet: Dromund Kaas

• Imperial Agent - Cleaning Up the Mess
• Sith Inquisitor – In The Halls Of Traitors
• Sith Warrior – Leaving Dromund Kaas
• Bounty Hunter – Joy Ride

Planet: Coruscant

• Jedi Consular - Ancient Secrets
• Trooper – Assault On The Works
• Smuggler – Race the Devil
• Jedi Knight – Race to the Ruins

The ship you receive depends on your class:

Jedi Knight & Jedi Consular – Corellian Defender

Smuggler – Corellian XS Stock Light Freighter

Trooper – Rendili Hyperworks BT-7 Thunderclap

Sith Warror & Sith Inquisitor – Fury-Class Imperial Interceptor

Imperial Agent – X-70b-Phantom Prototype

Bounty Hunter – D5-Mantis

Source and official site

Solution 2:

You get your ship when your class story arc takes you off the second planet (Dromund Kass or Coruscant). It's not directly related to your level, but I think you should be usually around level 15 at that point.