In many microelectronic systems, power rails are provided by DC-DC converters that have their own current limiting functionality. A reasonably featured LDO or buck converters, for example, will usually include a fixed current limit. More highly featured regulators might have a programmable current limit threshold.
For power sources that don't have a built-in current limit (a battery, for example), a fuse or re-settable fuse is typically used. In some situations, more sophisticated solutions using dedicated protection controller IC's may be employed. You can find these types of ICs marketed as surge stoppers, overcurrent controllers, circuit breaker controllers, etc. However, this approach would not usually be required for low-power loads like yours.
In short, consider an LDO with an appropriate current limit for your situation. You can run the LDO in drop-out so that it eats up only a tiny portion of your supply