City and towns in Massachusetts keep their own vital records but have sent transcripts or copies to the State since 1841.

• For records after 1906 - see Massachusetts Department of Public Health - Registry of Vital records and Statistics.
  
  
• For records between 1841 and 1905 - see the Massachusetts State Archives

  The records from 1841 to 1903 are in register-type bound books.
  Starting in 1903 death records became individual certificates and in the 1920s birth and marriage records also became individual certificates.

  Index to Massachusetts Vital Records 1841-1910 searchable database online at the Massachusetts State Archives.
  
  
• For pre-1841 records try the city clerk's office:
  

  Medford City Clerk's Office
  85 George P Hassett Dr
  Medford, MA 02155-3256
  (781) 393-2425
  

Other alternatives:
Massachusetts Vital Records to 1850 (microform)
compiled by Jay Mack Holbrook "Filmed from original town records" - should include Medford 1673-1849
Boston Public Library   Microtext Dept.  call #  F63.M485 1982x

Vital records of Medford Massachusetts, to the year 1850 (book)
Publisher : Boston, Mass., Pub. by the New England historic genealogical society; at the charge of the Eddy town record fund, 1907. Available at the Medford Public Library.

Vital Records of Medford to the year 1849 transcribed and indexed

Vital Records of Medford Massachusetts: to the year 1850 on Google books

Vital Records of Medford to the year 1850 online book at the Internet Archive Text Archives- American Libraries.

Mass. Genealogical Research is a private business that will do research for a fee, but they have an excellent information page on obtaining Massachusetts Vital Records for yourself. (This is not an endorsement of their services.)