If you are referring to "producer gas" or "wood gas", which is made primarily by the pyrolysis of biomass, it is composed of mostly hydrogen, CO, some methane, and similar, & H2O vapor, with some possible impurities like SO2, etc.

That is going to probably be impractical to store, and may be more dangerous to bother with.  The sulfur dioxide is also a corrosion contributor that would need to be scrubbed, although that can be done by running it through a chamber with steel wool, scrap iron, and similar that would be sacrificial, and replaced every few months or as needed.  The water vapor can be condensed out, but which would be worse?  A carbon monoxide leak, or a hydrogen leak?

Bio-gas like what you might get from anaerobic digestion of biomass, is going to be pretty close to what you have from a standard utility feed of "natural gas", although it is likely to have a much higher CO2 concentration.  That can be scrubbed by running it though a water chamber that was several feet tall (for pressure), and the CO2 will be absorbed into the water.  I've seen some details of doing it that way, but don't have any links on hand to share right now.  That would get you to nearly all methane, with a small percentage of butane and propane, but you will still have an issue of how to compress it for storage.  It may also need to be scrubbed of sulfur dioxide to prevent corrosion in downstream equipment.

Methane compression equipment isn't cheap, mostly because of the safety factors that are required to prevent leaks, and the high pressures involved.  Not exactly something for the budding amateur to start out doing...

If the ultimate goal is to heat a greenhouse, it may be more practical to just use the biomass as compost directly?  That's what they do at Growing Power, up in Milwaukee, and I've visited them to see it first hand in February, and it does work.

(http://www.growingpower.org/)  They also do a pretty serious aquaculture setup there too.