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, and New Hampshire offers same-sex marriage beginning in January 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, domestic partnership creates 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). Connecticut, New Hampshire, and Vermont offer civil union registration for same-sex couples, who can register their partnership and receive all the benefits of state laws that apply to married couples. The District of Columbia, Nevada, New Jersey, Oregon, and Washington State have domestic partnership that also is equivalent to marriage. Colorado, Hawaii, and Maine 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).