My Honda accord (+200 000 miles) starts with no problems in the morning but then if you turn it off after driving for while it wont start and wont crank, any ideas?
If the humming noise is more like a whirring sound, you have a bad starter-motor solonoid or "bendix."
First - open the hood and have one of you listen while the other tries the starter. If the starter motor makes a whirring sound but nothing else happens, it is certainly the "bendix."
The same switch that turns the motor on, also engages an electro-magnet (the bendix) that throws the starter motor gear out and engages the teeth in the motors flywheel thus allowing the electric starter motor to turn the engine over. When the engine starts, it spins the starter motor fast enough that it mechanically throws the starter gear back into the starter even before you release the starter and disengage the magnet.
OK - now that you know what it is, what do you do about it.
While the bendix can sometimes be cleaned and work a while longer, sometimes a really long time longer, it is seldom worth the effort to remove the starter mechanism, take it apart, clean and lubricate (with the proper lubricants) and re-install just to see if that will work.
Usually, the best solution is a new or rebuilt starter. You can get one on Amazon, Autozone, NAPA, etc. and the price will likely vary a lot. You can get one from the dealer as well but it may cost more than the car (some exaggeration). Check prices - they vary a lot, sometimes by as much as 2-3x. I think the original starter motor on your car is a Denso, but there are lots of good aftermarket starters available.
Starters are pretty easy to change. I suggest looking on youtube. Just make absolutely sure you disconnect your battery before starting this.
Changing the starter motor (the bendix is attached and comes with a rebuilt one) takes about 30 minutes or so on that car.
Basically, you disconnect the wires, unbolt the old one, slide it out, slide in the new one, bolt in in, and reconnect the wires. - then reconnect the battery and start the car.
Good luck.