My 24V motor's measured voltage was 8V when I connected it to a 24V supply
I recently bought a 24V 250W DC motor. ...


From \$ P = \frac {V^2}{R} \$ we get \$ R = \frac {V^2}{P} = 2.3 \ \Omega \$.


  In order to test it, I connected it to a DC power supply (rated +-25V / 1A). I set the PSU to 24V and connected the motor (no load) to it. The current immediately reached its cap of 1A and the motor voltage raised and capped at 9.3 V.


This indicates that the PSU has current limiting built in. Since we now know the resistance of the motor we can calculate the current supplied by the PSU as \$ I = \frac {V}{R} = \frac {9.3}{2.3} = 4 \ \mathrm A \$. The 1 A limit on the power supply refers to the maximum current where it can maintain 24 V out.


  I suspect that the current capacity had something to do with it because the motor did not get its ideal starting current (~10A). 


Correct.


  Is this normal behavior of motors (that the voltage is dependent on the amount of current I can supply to it)? If so, why?


It depends on a few factors. Most DC motors would start on reduced voltage if mechanically unloaded. You haven't given enough details to suggest why yours did not start.


  This will help me determine the battery I will use for my motor since batteries have recommended currents.


Correct.

Be aware that as the motor speeds up that it generates a back-EMF which opposes the driving voltage. This has the effect of reducing current draw from the supply and at light loads the motor run close to rated speed with low current. You may find my answer to Why does a DC motor spin when there is no load applied to it? helps you understanding a little.