Under the IP 4 standard, in place for several decades, the length of an address is 32 bits. There are about 4 billion such addresses, or a little more than 1 for every 2 persons on earth.
The IP 6 standard calls for 128 bits per address, or a total of 3.4 E38 possible addresses. If the earth has a radius of 6,380 km, then its surface area is about
4πr2 ≈ 12.6 * 40.7 E12 m2 ≈ 5.11 E14 m2 = 5.11 E26 μm2 = 5.11 E38 nm2
That's a little less than one address per square nanometer on the earth's surface.
Suppose that the carrying capacity of the earth is 10 billion people. If each person were to purchase one million new devices requiring an IP address every second of his or her life, then to exhaust all available addresses it would take
3.4 E38 addresses * (1 person-second/1 E6 addresses) * (1/1 E10 people) * (1 hour/3600 seconds) * (1 day/24 hours) * (1 year/365 days) = 3.4 E19/(3.6 * 24 * 365) years ≈ 1.08 E15 years, or about a trillion millenia.
By comparison, the observable universe is said to be about 14 billion years old, and the earth itself is only about 4.5 billion years old.