cx-270 or gx-270? I don't know of a cx series from dell but the gx series of optiplex machines were quite common, and a GX270 would be quite old. Regardless my first step with an older dell would be to open it up and look at the capacitors on the motherboard around the CPU socket area, dell got a bad batch back when GX-270s were being made that had a very high failure rate, if you see brown goo oozing out the top of the capactors or the tops look like they are bulging your board is toasted.
Next, we know that your PSU is putting out some power since your CPU fan is kicking up to full speed so its not totally dead. I would try reseating most of the cables in the case, if the CPU fan connector comes loose so that it doesn't report a speed the system wont boot but it will still try to power the fan.
If neither of those two are it you can move into some more elaborate troubleshooting, but from the sounds of it is it just an old system that may have reached its end.
I had an old 7 year old dell that died 6 months ago. The light in the front would turn yellow, but I wouldn't hear the processor like you do, the system wouldn't turn on. I later found that one of the capacitors on the motherboard looked to have been spoiled or just broke from the bottom. I tested the power supply and it was in working condition.
Sometimes, what works is to re-seat the RAM, processor, and all the cables and try again, that's a trick used for old dell computers.
By the way, my dell gave me these yellow light problems before, but it would turn on after 20 minutes. My yellow light problems happened after a power outage from a lightning strike, while the pc was on. One day I turned it off to test some case fans and it never again turned on. I guess the broken capacitors on the motherboard just finally gave out. Replacing the motherboard, etc. on the computer wasn't worth it.