Membranes are specifically designed by lipids to maintain internal hydrophilic environment in narrow range.There are hydrophobic amino acids among naturally occurring 20 amino acids and as well as there are also number of water insoluble polysaccharides like cellulose. These hydrophobic and water insoluble materials can also be used to build membranes but why not? Why lipids are only choice?
Membranes are built from a specific class of lipids, namely phospholipids, whose key property is that they are amphiphilic and so can self-organise to form bilayers. Not all amphiphiles do this, some prefer to adopt a micellar organisation. A bilayer composed of phospholipids produces the ideal combination of a hydrophobic barrier with a hydrophilic surface.
Although there are clearly amino acids with hydrophobic side chains, it is difficult to conceive of a proteinaceous barrier that would reproduce the properties of a phospholipid bilayer since it would not have a continuous hydrophobic phase.
Cellulose may be insoluble, but is full of polar groups so again would not create a hydrophobic barrier - I'm sure that ions can diffuse through a cellulose structure for example.
@WYSIWYG raises an excellent point in the comments. Archeal membranes are composed of a different class of amphiphile in which the hydrocarbon tails (branched; derived from isoprenoids) are attached to glycerol by an ether linkage. It's instructive to note that the overall design principles are retained however, suggesting that the self-organising properties of these molecules was the key to the "invention" of biological membranes.