Yes
First, you can arrange the frequency range to be used solely by distress signals, and arrange a set of omnidirectional listeners to detect the signal in that frequency range, to minimize unwanted interference from common broadcast. Next, while the signal emitter would also be an omnidirectional, you might feasibly arrange the situation on a distressed ship to allow a transmitting antenna to focus the signal at least into a single hemisphere, towards the star, both lowering the required power to be heard, and protecting the ship from the signal itself. In fact, ships on Earth have buoys designed to emit SOS and remain afloat without relying on the ship, your spaceships should be equipped with similar devices powered by say an RTG, so that they would have power to detach from a distressed ship, compensate its rotation, position itself between the ship and the star, align speeds with the ship, aim towards the star (then aim at a rescue base, if desired) and then transmit SOS with a small amount of data received from the ship (current course, local time, expected destination, flight number, problem code, etc - this can be compacted into 1 kB together with ECC) until deactivated.
Now, for the required signal power. Voyager 1's transmitter has "mere" 23W of signal power, and is heard from more than 150 AU away. Given that your "star system" range does not exceed 100AU (and usually the planets of interest are relatively lose to the star, give or take several AU), you don't need to broadcast your signals too far. With Voyager's antenna saying to have a 48 dBi gain, which translates into 10^4.8 or 63000x power (with 1/63000 area of full sphere), your omnidirectional or one-sided transmitter with 23W power would be heard from a 250 times less distance, or 0.8 AU, at the same power level as Voyager's, which is "barely hearable". If you employ a simpler antenna that can be directed and has a gain of ~20 dBi and aim it to the location of the nearest known (or just "known") SOS receiver, you can extend that distance tenfold, which would be 8 AU and quite viable to both travel in the first place, and be heard in the second place. Then, emitting a four times stronger signal would allow its detection from double the distance, although shortening the sending time by a factor of 4. An example proposed emergency buoy should be hearable in an omnidirectional mode from ~4 AU, expecting its RTG power of 4 kW, and ~40 AU if it succeeds to aim properly and use the directional antenna to send SOS.
Now for the reaction and signal time. With a payload of 1 kB, and current downlink speed of Voyager 1 of 160 bps, the signal's length should be longer than 50 seconds - it's a lot for an emergency, but since we're in SPAAACE, we don't care about length but care about being heard. I'd design several standard baud rates for SOS beacons, based on distance from the star measured by luminosity, or rather by the ship route's largest distance from rescue stations, and ensure that the ship captains would program their SOS beasons prior to leaving the docks according to their declared route and destination. This way the receivers could be more precisely filtering out random noise and better detect distress calls, if any,effectively making a SOS be heard from farther away. So, the minimum signal length is effectively dictated by whatever SOS protocols are running in your star system, and the distance between transmitter and receiver, the farther, the longer.
In total: Yes, the idea itself is feasible; your ship that falls into distress can send a SOS so that it'll be heard at the "base", that SOS can also have a payload, the minimum length of the SOS is determined by payload, distance and other predefined constants, but will be longer than 0.1s per AU^2, give or take an OOM, with currently used receiver of Voyager 1's signal. Using a worse receiver would proportionately lower the maximum detection range, but if there would be a space base, they will likely have a receiver of similar capability because of enough space around themselves, and low amount of unexpected emission.