Failed star/run capacitors are often bulged, sometimes even ruptured and/or have blackened spots around the leads. May be able to see that by just looking at the outside unit, sometimes even through the blades of the fan, but often need to take blades and the fan guard off.
Assuming you've checked the breaker/ contactor of the unit to eliminate something simple like a tripped breaker.
Anyway, getting back to the capacitor, if there's something wrong with it, replacement is the only option. Take the reading from the old one, check in hardware stores or online for the exact match or slightly higher capacitance/voltage. Never lower voltage but lower capacitance may work sometimes if you cannot find the exact replacement.
Failed cap is only one of the range of possible issues. Does the outside unit make any noises at all, any part of it turns (fan, compressor) or generates heat? Last but not least: what is the manufacturer and the model of the unit?