I'm going to go about this a little differently and say that the most unbreakable sports record is NFL quarterback Brett Favre's 336 interceptions. Let's look at the reasons why this record will never be broken:
In today's NFL, the quarterback position is the most important and scrutinized on the roster.
Long gone are the days where a defense or coaching genius could easily get you to the Super Bowl and a shot at the Lombardi Trophy. Over the past 10 Super Bowls (including this upcoming one), there have only been three out of 20 competing teams that arguably didn't get to the big show due to the exceptional work of their quarterback (QB): the 2002 Tampa Bay Buccaneers, the 2003 Carolina Panthers, and the 2006 Chicago Bears. Furthermore, only the Buccaneers came out victorious; the Panthers and the Bears respectively fell to teams led by Tom Brady and Peyton Manning, the two most-praised QBs of this era.
While Brett Favre is undoubtedly one of the greatest QBs to ever play the game, he is unique in the NFL of today in that no team would gamble on a veteran QB that throws up so many interceptions with all of his many touchdowns. With the exception of injury, one bad season now (heck, maybe even just half of a season) almost always leads to a back-up role or a starting position that one is just keeping warm until the offseason.
The time frame for a rookie QB to prove himself is now smaller than ever.
It's already been shown that there aren't any starting QBs like Favre in the NFL today. Now it's time to show why there won't be another QB like Favre again. While a brilliant rookie season might buy you a little bit of time, a less-then-stellar surely starts the death watch (the two most significant examples of this being Brady Quinn and Matt Leinart, two former first-round picks who were benched in favor of other QBs after less than a regular season-full of games). As I've already emphasized above, a high ratio of touchdowns to interceptions is key to a QB keeping his starting position in the NFL. While a bad season or two mixed in with eight or nine amazing ones is acceptable, two consecutively bad seasons or just as many bad as good seasons is not. Favre had more than his fair share of poor seasonal performances; if drafted today, he probably wouldn't even have been given a shot past his mediocre rookie season with the Falcons.
Performance concerns aside, a quarterback with as many seasons and starts as Favre is a rarity.
Brett Favre holds numerous records, most of them positive. One of these positive records is his number of consecutive starts, 297, which comes out to roughly 18.5 seasons as a starting QB. The only current QB close to this number of consecutive starts is Peyton Manning with 208. While Manning will almost surely pass this record, he will also almost surely never reach Favre's number of interceptions. Should Manning surpass Favre's record by playing for six more seasons, he'd have to average 22 interceptions per season to come to Favre's number of 336. However, save for two seasons, Manning has never thrown more than 19 interceptions in a season. Such a slide in performance is unlikely not only because of Manning's abilities but also because of the high probability that he'd be pulled in favor of someone else should he regress that much.
The NFL of today is extremely risk-averse, especially when it comes to the position of QB. Consequently, relative rather than absolute numbers are the most used indications of someone's performance. High-reward, high-risk QBs like Favre are simply not favored anymore in the NFL, and I find it highly unlikely that they ever will be favored again.