RAW Yes, but RAI No
RAW, the Con checks come at the end of your turn, and they are not explicitly prevented by ending the chase. So if you choose to Dash twice while at 4 levels of exhaustion, it is possible that you can kill yourself if you fail both checks, as your turn is largely over and you have no opportunity to take back your decision or otherwise influence the result. At even one level of exhaustion, you are at disadvantage on these checks, meaning death is a real possibility.
RAI, you will "drop out" of the chase when you hit exhaustion level five after failing the first check. The "Chases" rules set up chases as a special situation with their own special rules (like combat) and repeatedly imply that chases end when you stop moving (emphases mine):
The chase ends when one side drops out or the quarry escapes. (Beginning a Chase)
A participant drops out of the chase if its exhaustion reaches level 5, since its speed becomes 0. (Running the Chase)
A chase ends when one side or the other stops, when the quarry escapes, or when the pursuers are close enough to their quarry to catch it. (Ending a Chase)
These quotes make it clear that the intent is for the special situation of chases (and the rules that govern them) to end the moment the pursued or pursuers stop.
Further, the consequences of chases largely go away at the end of the chase. For example, there is a limit to the number of times one can Dash during a chase, but it is easy to reset that limit.
During the chase, a participant can freely use the Dash action a number of times equal to 3 + its Constitution modifier. Each additional Dash action it takes during the chase requires the creature to succeed on a DC 10 Constitution check at the end of its turn or gain one level of exhaustion.
Consider this - a PC is being chased and they have used their allotment of free dashes. They duck into a building and successfully hide, ending the chase. A round later, they are discovered by the building occupants, and a new chase begins. Since this is a different chase, and since the Dash limit is set per chase, they begin with a fresh Dash allotment.
The Dash limit is more like Initiative, which resets in each combat, rather than other features and abilities in the game, which reset every short rest, long rest, or day.
If participants do acquire exhaustion during a chase:
A creature can remove the levels of exhaustion it gained during the
chase by finishing a short or long rest.
Normally, exhaustion requires a long rest, food, and drink to remove just one level, whereas ALL of the levels of exhaustion from a chase may be removed by a single short rest, indicating that while it uses the exhaustion mechanic to affect play, it is in some sense lighter or less powerful than true exhaustion and is meant to be more temporary.
Finally, note that avoiding exhaustion in a chase is by means of an ability check, rather than a saving throw. We know that ability checks test "a character's or monster's innate talent and training in an effort to overcome a challenge", whereas saving throws, such as are used when a character undertakes a forced march, are made "because your character or monster is at risk of harm." This implies that while it may be a challenge to keep running while temporarily exhausted, doing so does not put you at risk of actual harm.
Thus it appears that chase consequences are intended to be "softer" than normal, and are meant to largely apply during the chase itself. Given this, I believe RAI is that when someone who has Dashed twice while at exhaustion 4 fails their first check, they are immobile and the chase (and all its associated mechanics) ends, meaning the second check is never rolled and running oneself to death is not a possibility.