Old English ham "dwelling place, house, abode, fixed residence; estate; village; region, country," from Proto-Germanic *haimaz "home" (source also of Old Frisian hem "home, village," Old Norse heimr "residence, world," heima "home," Danish hjem, Middle Dutch heem, German heim "home," Gothic haims "village"), from PIE *(t)koimo-, suffixed form of root *tkei- "to settle, dwell, be home." As an adjective from 1550s. The old Germanic sense of "village" is preserved in place names and in hamlet.
'Home' in the full range and feeling of [Modern English] home is a conception that belongs distinctively to the word home and some of its Gmc. cognates and is not covered by any single word in most of the IE languages. [Buck]
Slang phrase make (oneself) at home "become comfortable in a place one does not live" dates from 1892 (at home "at one's ease" is from 1510s). To keep the home fires burning is a song title from 1914. To be nothing to write home about "unremarkable" is from 1907. Home movie is from 1919; home computer is from 1967. Home stretch (1841) is from horse racing (see stretch (n.)). Home economics as a school course first attested 1899; the phrase itself by 1879 (as "household management" is the original literal sense of economy, the phrase is etymologically redundant).
Home as the goal in a sport or game is from 1778. Home base in baseball attested by 1856; home plate by 1867. Home team in sports is from 1869; home field "grounds belonging to the local team" is from 1802 (the 1800 citation in OED 2nd ed. print is a date typo, as it refers to baseball in Spokane Falls). Home-field advantage attested from 1955.
town n.
Old English tun "enclosure, garden, field, yard; farm, manor; homestead, dwelling house, mansion;" later "group of houses, village, farm," from Proto-Germanic *tunaz, *tunan "fortified place" (source also of Old Saxon, Old Norse, Old Frisian tun "fence, hedge," Middle Dutch tuun "fence," Dutch tuin "garden," Old High German zun, German Zaun "fence, hedge"), an early borrowing from Celtic *dunon "hill, hill-fort" (source also of Old Irish dun, Welsh din "fortress, fortified place, camp," dinas "city," Gaulish-Latin -dunum in place names), from PIE *dhu-no- "enclosed, fortified place, hill-fort," from root *dheue- "to close, finish, come full circle" (see down (n.2)).
Meaning "inhabited place larger than a village" (mid-12c.) arose after the Norman conquest from the use of this word to correspond to French ville. The modern word is partially a generic term, applicable to cities of great size as well as places intermediate between a city and a village; such use is unusual, the only parallel is perhaps Latin oppidium, which occasionally was applied even to Rome or Athens (each of which was more properly an urbs).
First record of town hall is from late 15c. Town ball, version of baseball, is recorded from 1852. Town car (1907) originally was a motor car with an enclosed passenger compartment and open driver's seat. On the town "living the high life" is from 1712. Go to town "do (something) energetically" is first recorded 1933. Man about town "one constantly seen at public and private functions" is attested from 1734.