The number 600,000 is firmly embedded in the Jewish psyche. It is, of course, the (approximate) figure that the Torah gives to the number of adult (age 20 and over) Jewish males in both the census of Bamidbar chapters 1 and 2, and the later census of chapter 26. This number has been extended to various mystical ideas such as the number of root Jewish souls, the number of letters of the Torah, and the number of interpretations of Torah.
However, numerous objections have been raised to the 600,000 figure, based on such considerations as our current estimations of the population of the ancient world, the ability of the land of Goshen to support this number of people, such a huge number of people being able to cross the Yam Suf in any reasonable amount of time, the absense of evidence of such a large number of people entering into Canaan en masse, etc. An excellent discussion can be found in Colin Humphreys' The Miracles of Exodus: A Scientist's Discovery of the Extraordinary Natural Causes of the Biblical Stories. Chapter 8 - "How Many People Were In The Exodus" - goes into detail regarding both the problem and a solution based on interpreting eleph as "troop" instead of "one thousand".
There is another serious objection that I have not seen raised before.
Aryeh Kaplan's The Living Torah is a masterpiece. For those who think that it is merely a translation of the Torah into modern English, look again! The prodigious and well-researched footnotes that refer to a plethora of classical Jewish commentaries, plus the maps, tables, and charts make this a must-have for any Jewish library. But a curious thought arose when I looked at Plate 26 on "The Camp". I have added some notes to the chart which gives one a better understanding of a major problem that results when ones accepts a literal approach to 600,000 in conjunction with the asserted layout of the Israelite camp in the desert. (Click on the image to make it more legible.)
The problem? Population density! Up to now I've only mentioned the adult male population for each tribe. One typically sees a number of two to three million Jews in total (although based on the fecundity of Jewish women that is often invoked in midrashim, 1-2 children per family seems to be extremely conservative.) We'll use the commonly given figure of 2.5 million people living inside an area of 5 square miles. This gives each person less than 60 square feet of total space, space that also had to accommodate sleeping, eating utensils, etc. (For the purpose of this discussion we'll grant the fundamentalist various well-known miracles that obviated the need for latrines or areas for food storage. Animals would have been kept outside the habitable areas of the camp.) For comparison, this is 14 times the population density of the densest city in the world - Mumbai, India! And Mumbai, of course, can accommodate this density only because of the many tall apartment buildings; the Israelites all lived on the ground floor.
It would have been bad enough for the tribe of Manasseh, the least populous of the tribes with about 92 square feet per person (again, assuming a total population of adult males * 4) but Yehudah? Less than 40 square feet per person! (If you are an average height male of 5' 9", hold a one foot ruler in each hand grasping each one at the 4 inch mark, extend your arms to each side and rotate 360 degrees. That's how much personal space you would have.) Sometimes it just doesn't pay to be the BMS (Big Macher Shevet)!
Again, the population density would be even more preposterous if one were to use a more reasonable number of children per family - the 2.5 million number assumes that every male was married with only 2 kids! (Who did they think they were, Modern Orthodox??)
Jews have lived in extremely crowded conditions before. The Warsaw Ghetto is an unfortunate - but notable - example, but at its worst the maximum population density was only half that of the supposed desert camp population and the ghetto consisted of mostly multi-story buildings. Keep in mind that the ancient Israelites supposedly lived in such conditions for 40 years...
Regarding the proposed dimensions of the camp, Rabbi Kaplan refers to Rashi on Bamidbar 35:5, even though the latter passage speaks of the areas that the Leviim will inhabit once they conquer the land of Israel. Perhaps this was learned out by a gezerah shava (סָבִיב)? In any event, I haven't come across any other source that posits what the area of the camp was, so this is certainly not the strongest argument that can be made against the fundamentalist approach which insists on a literal interpretation of 600,000. It's more of a little chip off of the already crumbled edifice of Biblical literalism.