The capacitors size themselves are small. They slightly vary but are almost always the size of a medium Red Bull can. They are sized to your unit though so you can’t just buy one without looking at the specs and expect it to work. But the specs are printed on the side of the can and if not can be derived from the specs on the unit itself.

As far as their danger there really isn’t any beyond getting shocked from dealing with live wires. Technically they can retain a bit of a charge so I’ve seen recommendations to wait X amount of time before touching them with your bare hands or to discharge it by touching it with an insulated screwdriver to discharge it but the risk is pretty low. Once the power is off (either at the breaker or via the disconnect at the condenser unit, power only goes in one way to those things so if you turn it off in one place there’s no way you’ll get a zap) it’s a soda can with 3 wires going into it. You just disconnect the 3 wires from the old soda can, remove it, replace and connect the new one. Not that much harder than changing a light bulb.

the capacitors are connected across the motor windings and are there essentially as a way to shift the phase of the current waveform. note that this is a two-phase power special; you don't need start or run capacitors if you're using 3-phase power (uncommon in North American residential settings, but YMMV worldwide)

when power is disconnected they are not charged at all. it's not like the capacitors you might find in a CRT

My A/C failed in the same way and with some help from youtube and a multimeter I debugged it to the same problem. Replaced it with this one: https://www.amazon.com/gp/product/B00GSU47TQ

There's a bunch of similar capacitors on Amazon (or your local hardware store). They're about the size of a soda can. I believe the "old" capacitor in your A/C can zap you if you don't ground the lines together when you pull it out, if you watch youtube videos for this repair they'll ground it with a screwdriver or other metal object.

Turn off the power or pull the local disconnect. Also, short out the capacitor before touching it. You can do that by connecting the terminals with your screwdriver. There are plenty of yt vids explaining the process.

When I see large capacitors (for me it’s more girth than my little finger), I have alarm bells going on. One « soda can sized » can definitely kill you. It should be discharged before messing with it. You can buy capacitor drains (basically a big ass resistor) that you put across the capacitor’s legs to drain the energy in it.

Some do it with an insulated screwdriver but that’s dangerous because it’s a short, can ark, fuse the driver to the capacitor, and result in a bad day.

to be safe short it with a screwdriver (unplug the unit before), unlike a lion battery it won't explode.

it's recommended to give it some time if it's been running and to short across the terminals with something, like a screwdriver that has a non conducting handle. It's nothing too ridiculous.