Yes. In May 2004, Massachusetts began issuing marriage licenses to same-sex couples who live in the state, and California allowed same-sex marriages from June through November 2008. Connecticut begain issuing marriage licenses to same-sex couples in November 2008, and Iowa did the same in April 2009. Vermont followed suit in September 2009, New Hampshire in January 2010, and the District of Columbia in March 2010. Many other states have passed laws specifically barring same-sex marriages, and the number of states with such laws is increasing. However, there are states that allow same-sex unions that are similar to marriage.
In California, Nevada, New Jersey, Oregon, and Washington, domestic partnership or civil unions create a marriage-like relationship in which same-sex partners have nearly all the rights and responsibilities of spouses (and partners who are married may also register as domestic partners in California if they wish). Colorado, Hawaii, Maine, Rhode Island and Wisconsin all have some form of registration for same-sex couples, with varying levels of benefits -- but none are marriage equivalents.
For more information, see Nolo's article Same-Sex Marriage: Developments in the Law. For guidance on whether to enter into a marriage or other legal relationship with your same-sex partner, see Making It Legal: A Guide to Same-Sex Marriage, Domestic Partnerships & Civil Unions, by Frederick Hertz with Emily Doskow (Nolo).