The 五段 verb 堪る means "to endure". In this lesson, we will see it used after the て form to show the speaker's utter intolerance for something.
～て堪る is followed most frequently by either か or ものか to show that one would never let something happen. The speaker is showing a strong resolution of not having something be in such a state/condition. As you can imagine, this isn't used in polite speech.
This is synonymous with 断じて～など（し）ない. 断じて ≒ 決して. ～ものか shows a strong sense of negation, and it can also be seen in more casual speech as ～もんか. It literally creates a strong negative rhetorical question.
I would never fail!
I would never be defeated!
I would never die!
I would never allow this.
Sentence Note: 4a shows the speaker's incapability of not allowing such. 4b shows a strong negative emotional reaction of allowing such, and the latter as with all of the other sentences with ～てたまる（もの）か invoke emotions such as anger. So, although the sentence with 断じて could be reworded to be even polite, that's impossible with ～てたまるものか. It's a phrase that should be limited to one's ウチ.
たまらない can be translated as "to die for", "ache", "kill for", etc. You literally can't stand it, and as you see below, this is often used in a positive sense. When it's not, it should be obvious by what's being used. For instance, in the first example, the speakers really want to go, and you get the sense that they just can't stand waiting any longer. This phrase is sometimes seen in polite speech as Ex. 4 illustrates.
We are eager to go.
I'm dying for a break from the heat.
I couldn't stand not having a drink.
She is irresistibly beautiful tonight.
To get furiously mad.
I'm aching to travel.
I can't stand but regret losing.
I can't stand losing.
1. Although the subject of the last sentence could be different, both sentences show the speaker's/subject's not standing the fact of losing. The positive/negative associations of this pattern clearly fall on semantic lines of what you're using in combination with it.
2. わけ ＝ Reason. いかない ＝ Won't go. So, ～わけにはいかない literally means "won't go with the reason of...". It can be after positive and negative sentences. So, just like in English, a double negative in this situation would make a positive.