2001 Elantra AC question

2001 Elantra with ~31K miles

When I turn on the AC you can hear the compressor kick on but the air
doesn't get colder. If I keep the AC you can hear what sounds like
the compressor start again and again, but still no cold air.

Would you guess that I just need to charge the system or do you think
this is a bigger issue?

 

You have GOT to assume you have some kind of issue here.

Especially with a car of the 21st century (even if only barely), I cannot
imagine being able to "just (re-)charge a system," without going through a
system and leak test.

The 134a had to go somewhere. The days of cars just needing to be recharged
once a year are long past.

 

The open grill on the Elantra makes it prone to A/C condenser damage
from rocks thrown up by other vehicles. It's a pretty common problem.
Many Elantra owners (like me) have taken to covering the inside of the
grill with Gutter Guard plastic mesh to prevent rock damage. There are
DIY's on the subject on the Elantra Club site.

While it's possible that your issue is a reparable leak, brace yourself
for the possibility of replacing the condenser.