The answer is 'yes' and 'no'. Let me explain.
As Bill mentioned, there are plenty of implementations of various OGC standards WFS, WMS, TMS, etc. (see Geoserver, Mapserver, Tilestache, etc).
However, I would guess that when some of your customers ask for this, they may mean ESRI ArcGIS Server REST Endpoints with ESRI-specific services. The answer as of today - is no (although I have seen some people implement successfully very small subsets of these endpoints in their favorite programming language).
You see, sometime back, ESRI was pushing to make these ESRI-specific endpoints an OGC standard which would have caused various open source projects to implement them. Nevertheless, it was a super heated debate since, among other things, was interpreted as having one vendor trying to push a "standard" that competed with other very well defined, largely implemented, OGC standards (e.g. WFS and WMS) that were in use just for the sake of having the vendor put an "OGC approved" label.
So yes, ArcGIS server supports REST endpoints that implement OGC standards and you will find open source implementations of those... and no you will not find complete open source implementations (as of today) of the ESRI-specific ones.