Yesterday, our A/C shut off and would not come back on. It shows on the digital thermostat that the fan is blowing cool (two different icons that appear and move) but the unit is NOT on.
After leaving it off for the afternoon, my husband was able to turn it back on, but this lasted 30 seconds and then it shut off again.
I tried again this morning, after having it off all night with the same results. Thirty seconds and it's off.
What could this be and can we fix it ourselves?

Are you saying it's not blowing any air out of the vents or is it just the outside unit(Condenser) that quits? If it's the fan that not working, the you probably have a defective blower motor that's heating up and going out on overload. When it cools down(sits for awhile) it will restart and then go out again. Not real uncommon.