An enclosure may be provided with a housing. The housing may have an opening that separates and joins along a seam. A magnetic closure may have a pair of mating flexible magnetic members such as flexible rings or strips that run along the seam. Magnetic-field shunts may be provided in the flexible magnetic members. Multiple magnetic elements may be formed along the length of each magnetic member to form a multipole permanent magnet. Magnetic attraction may be varied along the lengths of the magnetic members and may be varied as a function of the separation between opposing magnet members. A magnetic clasp may be provided on the housing. The magnetic clasp may have attracting magnets to help close the opening in the housing. Protruding clasp portions may be formed from flexible material and may have embedded magnets configured to repel each other to facilitate use of the clasp.