In the most recent chapter of Fairy Tail, Chapter 400:
It was revealed that Igneel was residing inside Natsu, OR was using Natsu as a portal from the place where he was residing to come to Earthland. If Natsu had such magic imbued into him that could cause interference with the barrier as logically speaking, the barrier prevents those of age 80+ to leave and if it's a portal for someone of age 80+ to leave the barrier would automatically stop that person from leaving OR Igneel was actually inside Natsu not just using him as a medium and that would most definitely stop Natsu and Gajeel from leaving the barrier because they have someone of at least 300 Years of age in them. Since dragons are extinct, it would be safe to say Igneel and the other dragons are at least that age.
There have been many theories on this subject, in fact. Here are some that I believe are logical and ones that might help:
Natsu and Gajeel's Dragon Slayer Magic is very ancient, obviously over 80 years old. Maybe their magic's age prohibited them? (Personally, I feel that something's missing here.
Natsu and Gajeel (and Wendy as well) were raised by their dragons a long time ago- maybe over 80 years ago. Through the Eclipse Gate, they were transported to the future- July 7, year x777 by Layla Heartfilia (Lucy's mother), making it seem like their dragons disappeared as opposed to them leaving their dragons. (Of course, there are a lot of pieces missing in this explanation, but it seems logical if you think about it.)
Laxus knew that Natsu and Gajeel were incredibly powerful, or maybe he knew that the Dragon Slayer magic is very powerful. There's a chance that he didn't want them participating due to this and told Freed to incorporate this into the rules without telling anyone about it. (Maybe?)
Anyhow, I hope this helped, and I hope that Mashima reveals this in the future!!
Until now, it was never explained properly in FT.
Future chapter might give a proof of the reason, or it stays a plothole.
There are some theories, like "Dragons are extinct nowadays, Natsu was raised by Igneel hundreds of years ago and made a time travel (similar for Gajeel)" or "Dragon slayers are real dragon children (with longer lifespan than humans) with some kind of transformation magic applied" (this conflicts with Zirconis´ story, but it´s still possible he didn´t know or lied. Mashima likes trolling anyways)
I believe the rules were not all stated. This is somewhat a trick from Laxus. Laxus told Freed to prohibit Dragon Slayers from entering the fight and to trick them he told freed not to put the prohibition of the Dragon Slayers on the rule list.
Based on Fairy Tail chapter 436, it was somewhat explained why Natsu can't past the barrier.
Natsu's physical body is more than 80 years old. Zeref used the body of his little brother to create Natsu and Zeref himself is about 400 years old. This is the likely answer why he can't past Freed's rune.
Reason for why Gajeel can't too has not been explained as the current chapter.
For Natsu, on one episode Igneel did tell Natsu that he was always in Natsu's body and that tell us that well Igneel is definitely older than 80 years old and if he is in Natsu's body, Natsu's age would definitely be Igneel's age and same thing with Gajeel
The reason is that the dragons are sealed inside of hem. Their dragons are over 80 years old. The enchantment is not allowing the dragons to go through, not Natsu and gajeel. But since they are combined, that’s why they can’t go through either.
natsu like all of the other dragon slayers, aside from laxus and cobra were born over 400 years before hand, but this isn't the reason natsu and gajeel were not able to leave. its because they had dragons in them who were atleast 80. dragons are immortal. so they don't age like humans. but natsu, gajeel and the others are not chronologically 400 years old. as chronological refers to a measurement of ones existence inside a flow of time. since they time traveled they are not 400 years old. the gate opened on one side and they entered it. while in the gate they were outside of time for 400 years. layla opened the gate in 777 which allowed them to leave. notice how zeref wasn't able to travel backwards in time using the ellipse gate, its likely because its not traditional time travel. its more like they left the flow of time and sat outside of time and space for those 400 years and when layla opened it they they could enter the modern world. so its akin to a stasis field where they are literally outside of time. zeref needed not only fairy heart but the time rift to travel backwards in time. this also creates a plot hole with future lucy and rogue going backwards in time. but that could easily be explained a way by the fact that future rogue and lucy came from alternate timeliens where the gate worked differently
I have a theory that you know how zeref is over 400 years old but if you look at the episode where natsu sees zeref zeref cries tears of joy saying saying you`ve grown natsu and we meet again so I think that zeref might actually be related to natsu and since zeref is 400+ maybe natsu is 100 and over since ive heard that zeref had and army of dragons maybe igneel and the other dragons slowed down time to where natsu and gajeel may look 17 to 18 but is probaly older than that but this is just a theory who knows im probaly right