You will need a member of your community who has the Sheriff leadership style. To find out someone's leadership style, go to the community screen, select him or her, and press
to the "Standing" box. Once he or she has accrued enough standing to become a Hero, you can make them your Leader. Generally speaking, standing is earned by taking control of that survivor and using him/her to complete objectives. Do that.
You will also need a home base that has a large outdoor slot. When you have your Sheriff Leader and your large outdoor slot, simply gather the materials required for the Field Hospital, build it, and wait. When the facility is finished, the achievement will unlock.