There's work towards this, from the X-ray emission of pulsars[0]. I don't think you'd need a precise distance to the pulsars to measure your own position, as long as you're measuring your own position relative to some other object (e.g., Solar System barycenter). If it's a relative measurement then you should be able to determine that by comparing the time-of-arrival (TOA) of the pulsed emission from multiple pulsars with the TOAs at Earth.
[0] https://en.wikipedia.org/wiki/X-ray_pulsar-based_navigation