The voltage handling capacity of the wire is related to its insulation, not the conductor size. You can safely put 24V (or even up to 48V) on typical telephone wire. The telephone company does it all the time.
The current handling is related to how much voltage drop you can tolerate as well as how hot you're willing to let the wire get. Voltage drop is influenced by the length of the wire while heating is to a much lesser extent (the longer it is, the more power it will dissipate, but the larger the space it has to dissipate it over). As a total WAG, I would say AWG24 is probably fine to about 2A over short distances for chassis wiring type applications.
So, if you need 70 LEDs, each 3.3V @ 20mA, you might find the following configurations work well:
For a 12V supply, 3LEDs in series by 23 strings = ~10V on the LEDs (R = 2/0.02 = 100ohms). Supply needs to handle 23*20mA = 460mA. Not too bad.
For a 24V supply, 7LEDs in series by 10 strings = ~23V on the LEDs (R = 1/0.02 = 50ohms). Supply needs to handle 10*20mA = 200mA. Easy peasy.
You could go to 48V, but there's not a ton of reason to.
For these solutions, the worst current seen (that coming off the supply in total) is well within what you can run some distance over typical AWG24 telephone cable.
If you only have 5V at your disposal, you have to run them completely in parallel. That's 70*20mA = 1.4A on the supply, which is modestly hefty for a typical wall wart, but not a huge deal (a typical powered USB hub has a 5V@2A supply). Your resistors each dissipate 1.7*.02=34mW for a total of 2.38W wasted, whereas the 48V solution only wastes 200mW across all 10 resistors (20mW each).
In general, even the 5V solution is manageable. You'd probably want to either double- or triple-up the wiring from the supply to the LED group or run each LED's wiring straight back to the supply.
EDIT: Erm, I did the math up there for 70 LEDs, not 73. Same idea, though. If you have an "odd string out", you just re-size the resistor on it. If it's substantially off, like 3 left vs. 7 on a "typical" string, you can run a few strings of 6 to balance the power back out so you don't end up with one big, hot resistor on the short string.