Most open source GIS programs can open all most common, and quite a few obscure data format (thanks to the gdal library). What they can’t open is the ArcMap layer files (.lyr). A layer file defines the symbology used to draw the data when it is added to the display and stores information about the symbology, symbology classifications, labeling properties, scale dependency and definition. So what if a data layer is distributed with a layer file?
Well, most likely you’ll have to recreate the legend yourself. A bigger problem is if the data is distributed with the key attributes described in the .lyr file rather then in the attribute table. In other words, if it contains the classification of the layer.
A good example is the Terrestrial Ecoregions of the World data set distributed by WWF. It is available as a shapefile with two .lyr layer files that can be used to symbolize respectively the ecoregions and biomes. The layer file is indispensable; the attribute table only contains a numerical code for the biomes, while the layer file contains he corresponding classification. So without ArcGIS, you will need to recreate the legend and to somehow find out which number in the attribute table correspond to which biome.
If you happen to have access to a computer with ArcGIS, there is a tool that helps you to convert the layer file to a .sld file, ArcMap2SLD. It is a tool for reading out styling information from ArcMap and converting into Styled Layer Descriptor-Files (SLD). This is a OGC (Open Geospatial Consortium) standard for web map-service (WMS) symbolization. It is also a standard that can be used by QGIS.
Not an ideal solution, you still need access to ArcMap, but at least it makes it possible to convert ArcGIS layer files. It also means there is not really an excuse anymore to distribute your data with .lyr files only (and i.m.h.o. , it is always a bad practise to describe key map attributes in the legend file rather then in the attribute table).
And assuming WWF doesn’t mind, you can download two above-mentioned legend files in SLD format here.