Dell Vostro 1500 making loud beeping noises and locking up

It probably is bad memory or a problem with the CPU fan. If there's multiple memory sticks, try it with only one and switch it around between the slots.

How long can it stay on?  What is the beep code?   Do you have Mem86 to run on the machine?  How old is this machine and are all of the fans running?   I have worked with some Dell if the side fan has a bunch of dust it in it will cause it to freeze and lockup.  I would checked there first

the machine is about 3 years old.  it stays on for anywhere from a couple hours to 20 minutes before it experiences any problems.

i'll look into both the fan and memory.  any suggestions for diagnostics that i can try?

Try running mem86 or the dell diags

it can be overheating; check cpu , disk and video gpu temperature with speedfan :
http://www.almico.com/speedfan.php      
if it is overheating, clean out dust, or replace the heatpaste on the cpu !

ran diagnostics at boot and it returned error code 0146, 2000-0146, with a message that DST shortest fail.

I did some searching and it seems like this is related to the hard drive.  Anyone have any other suggestions?

Replaced the disk and everything is well and fine.

tx for the feedback !