You sure can! Obviously, when you live in an apartment you will have restrictions on your ability to install radio antennas. Yet there are several steps you can take to enjoy ham radio. First, if your interest is primarily in local area communications, you can use VHF and UHF radio equipment. Often, you will only need a handheld radio to access a local voice or packet data radio repeater. Some apartment-bound radio operators install small beam antennas inside. I know one ham operator who operates amateur television equipment - literally transmitting color TV signals from his apartment. Fortunately for him, he had a window that opened up to face a nearby mountain peak that was home to an amateur radio TV repeater system. So his signals were then retransmitted over a very wide area enabling him to communicate using full motion, full color and sound television signals. If you wish to operate on the shortwave bands (known as "HF"), there are several alternatives you can consider:
(1) you can install a wire antenna on the ceiling of your apartment and operate at reduced power levels to avoid causing interference to neighbors caused by your antenna being close to their TV or sterio;
(2) install a dipole antenna for the higher HF bands in a balcony area;
(3) string an "invisible" wire antenna made from small gauge wire from your apartment to a nearby support such as a tree;
(4) use a closed tuned loop antenna mounted in a balcony;
(5) set up a good mobile radio installation for the HF bands and operate from your vehicle. All of these methods can prove very effective at providing HF communications for persons having limited space for antennas. I know of two amateurs who asked for and received permission to extend a single wire antenna across the roof of their apartment building and were thereafter able to enjoy HF amateur radio operation.