Many FAS employeees and postdocs live in the neighborhoods closest to the Cambridge campus, namely Cambridge, Arlington, Allston, Watertown, and Somerville, but obviously cost and transportation are significant limiting factors. 
Parking is expensive and hard to find in many neighborhoods. Parking on campus is also expensive. Many people instead choose to utilize Boston’s public transportation system--the MBTA, often called “the T.” In addition to being a more economical option (as well as often a time-‐saver), benefits-eligible appointments receive the option of a 50% subsidized “T-pass.”
Harvard offers a commuter bus for Harvard affiliates from Harvard Square to Longwood Medical School – the M2 shuttle. The M2 shuttle is free with your Harvard ID card. However, the service is infrequent after 7pm and on Saturdays. No service is offered on Sundays or holidays. A schedule can be found here.
Some people choose to bike or walk to work. Harvard offers many benefits for bike commuters, and bike racks and routes are available. A map including walking times and paths can be found here. Please note that in especially rainy months or winter months (where weather varies wildly from bitter cold to snow to sleet or other precipitation), biking or walking to work may be only for the hardy souls.
How to start looking:
Advice about housing can be found in the guide produced by HU's Advanced Leadership Initiative (ALI), although some of the options may be more relevant to ALI Fellows than postocs.
Harvard Off Campus Listings: After creating an account, you can search for apartments and find roommates.
Harvard University Housing: A convenient option for the close locations, online application process, no last month’s rent, security deposit, or finder’s fee, and maintenance of properties. HUH also programs a number of community engagement opportunities to all residents through the Graduate Commons Program. Depending on the time of year you start there may not be many options available, although some apartments typically become available throughout the year. We recommend that you start your housing search early so you can get a sense of the market.
If you are benefits-eligible, you have access to certain housing and mortgage resources. Click here for more information.
An incoming applicant’s webinar from the Harvard housing office can be found here.
Listings of apartments to rent can also be found on Harvie.
Some non-Harvard options (Please note that these housing resources are completely independent from HUH, and HUH does not endorse any of them or review the information given by these resources.) Please also beware of housing scams. Read this before proceeding: https://berify.com/blog/craigslist-rental-scams/
The Boston Posdoctoral Association has compiled a list of resources in its "Postdoc Housing Resource 2018."
Some Harvard and MIT facebook groups:
https://www.facebook.com/groups/gsdhousing/?ref=br_rs
https://www.facebook.com/groups/HarvardMITHousing/?ref=br_rs
https://www.facebook.com/groups/harvardgradmarket/
Airbnb
Airbnb is a short-term solution for several weeks or months. If you want to arrive and then take your time to find permanent housing, this may be a helpful solution.
Craigslist
Another good resource to find available apartments or roommates in the area.
In general, if you're wondering, according this Science Magazine article, cities offer great postdoc opportunities, but the cost is an issue.