Answer
The Bible is not very clear on what happened to Enoch (Gen. 5:24). It would appear that he was a very godly man, and that God was so pleased with him that the "took" (Hebrew laqach) him to heaven long before he normally would have died. He was 365 (Gen. 5:23), but most of his relatives mentioned in Seth's genealogy (Gen. 5:3-32) lived at least twice as long. Traditionally it has been understood that Enoch was taken bodily to heaven, but the Bible is not explicit on this point. What is explicit is that he did not die (Heb. 11:5).
Something similar happened to Elijah, though how similar is hard to say. In 2 Kings 2:9-13 we find an interesting account of Elijah's ascension into heaven. Like Enoch, Elijah was "taken" (Hebrew laqach) to heaven without having died (2 Kings 2:9-10). The similarity of the language with Genesis 5:24 is hard to miss, and it may indicate that Elijah and Enoch were taken in similar fashion -- not necessarily by whirlwinds and chariots of fire, but bodily and without dying.
We also know from Scripture that God created us to consist of both body and soul (1 Cor. 15). He did not intend us to be disembodied spirits forever. Also, death is an enemy (1 Cor. 15:26) that separates us from our bodies for a time. If we bypass death, there would seem to be nothing to kill our bodies, and therefore no curse of separation of body and soul.
In 1 Corinthians 15:51-54 Paul also teaches that when Christ returns some of us will still be alive. Without dying these people will receive new bodies just like those of the people who are resurrected to eternal bodily life. In this context Paul teaches that Christ is the first to have received his new body because he is the "first fruits" of those who have died (1 Cor. 15:20,23). Paul does not make mention of the exceptional situations of Enoch and Elijah, so it is hard to know how to apply his argument regarding new bodies to these two men. Hebrews 11:40, however, says that the saints who have gone before us are not perfected, not because the resurrection has not yet occurred, but rather because they are waiting for us. This might imply that they have yet to receive their permanent bodies, though it is not a necessary implication (e.g. they have already received greater blessings by being in heaven). Paul does say that Christ is the first to receive a resurrected body (1 Cor. 15:23), but I'm not confident that he intended his argument to rule out the possibility that Enoch and Elijah have bodies that are improved or changed in some way.
In summary, I think that Elijah and Enoch probably have bodies in heaven. While we may hypothesize regarding the nature of their bodies, however, we cannot be dogmatic on the issue.
Answer by Ra McLaughlin
Ra McLaughlin is Vice President of Finance and Administration at Third Millennium Ministries.