Could be your power supply acting up. To "reset" the PSU, disconnect the power cord from the computer itself. If the power supply has a physical toggle switch, push it to the off position. Then press the power button on the front to discharge any residual power built up in the system.
Then reverse the above and try again. See if that helps.
If you need a replacement power supply, txcesssurplus.com is a good source for older Dell parts.
Good luck!!!