“His eyes were small and close” is questionable, ie may be semantically incorrect. Faced with such a sentence, I'd interpret close in its sense of secretive, or “not willing to share information about yourself or your emotions”. See the last-listed adjectival sense in the right sidebar at onelook.com's entry for close.
“His eyes were small and close together” implies the distance between his eyes is less than usual. The sentence “His eyes were small and close-set” would be nearly equivalent.
“The two buildings are close” is what one says to say that two buildings are near the speaker or near some previously-mentioned reference point (although close by is likely to be used in the latter case: “The two buildings are close by.”).
“The two buildings are close together” means the two buildings are not far apart; which, depending on context, might mean the easement between them is inches instead of feet, or might mean they are on the same block, etc. In any case, the sentence doesn't say or imply whether the buildings are near the speaker or an aforementioned reference point.