Internet addresses - not email addresses but the unique identification numbers of Internet 'host' computers - are running out. But the shortage should be solved over the next few years as hosts gradually switch from the existing 32-bit addressing scheme to Internet Protocol version 6, which uses 128-bit addresses.
IPv6 is designed for a future where every VCR or toaster has its own address. Robert Hinden of Ipsilon Networks explains in this month's Communications of the ACM that IPv6 provides 655,570 million million million Internet addresses per square metre of the earth's surface. This should be plenty, even allowing for inefficiencies in the address allocation scheme.