I just remembered, we can also look at translations of English sentences that contain "cope" submitted on Tatoeba.org. (A list of all sentences is found here.) The various verbs used for this on Tatoeba are:

• alfronti, in the sentences "Kiel viaopinie Tom alfrontos la situacion?", "Li rapide alfrontis la situacion," "Ŝi ne kapablas alfronti streĉon," etc.
  
• devi superi, in the sentences "Vi devas superi tiujn malfacilajn problemojn," and "Vi devas superi tiujn malfacilajn problemojn."
  
• elteni, in the sentence "Ŝi ne eltenas streĉiĝon."
  
• sukcese lukti, in the sentence "Kiel ili sukcese luktos?"
  
• povi trakti, in the sentence "Neniu povas trakti lin."
  
• traktigi, in the sentence "Nunmomente, Tomo apenaŭ traktigas sian laborŝarĝon."
  
• majstri, in the sentence "...por majstri la proksimiĝantan senprecedence maljuniĝantan socion ĝis la sekva jarcento." (the full sentence is a little long)
  
• primajstri, in the sentence "Primajstri la kapricojn de la vetero estas esenca parto de la vivo de terkulturisto."

Rajzino: snip....
zich weten te redden - zich = oneself; weten = be able to; te redden = to save/rescue. So "to be able to save oneself".
"He coped well" would be "Hij wist zich goed te redden", ...

I don't think this bears an exact relation to cope as used in English. Redden (kun je het wel redden? etc..) is more like saying 'do you think you can make it?, but not 'do you think you can make it through? (a problem or issue. It's not about a situation of being able to manage under great duress or difficult circumstances. Coping is defined in dictionaries as simply 'dealing with' something successfully, but the connotations of its common English use are in circumstances like how a bereaved person manages feelings after a death or a bout of depression, or bringing up children in a bad situation, i.e when they struggle to have the will or the means.

Rajzino:And the two for "cope with" are:

...snip...

• bestrijden - meaning to fight, combat, resist, counteract, control (e.g. a plague). Which I think is not really the nuance we are looking for here.

Can anyone find or think of a better translation for taking sth./so. on? Because I think that might be on the right track.

These second ideas for 'cope with' are not much to do with the meaning of cope as 'managing'. 'Cope with' is not a separate meaning.

Alpaŝi (deal with, tackle) is the direct meaning of cope in English and can be woven into any sentence to express it. The special connotations it has acquired in English can't just be bolted onto a translation made to another language.

I suspect that the core meaning which the OP was looking for a translation for was that the situation is overwhelming for the person concerned. It is beyond their emotional, physical, financial, skill resources.

So one way to approach the translation might be to centre it on the situation rather than the person.

However there are quite a few uses of 'elporti' in the Tekstaro, where the meaning comes close to 'cope with'.

Pli granda estas mia puno, ol kiom mi povos elporti
La bramano ne povis elporti la turmentojn, ricevatajn de sia edzino, kaj foriris
Kvankam la tabuloj povis elporti mian pezon, ili ...
Li ne povis plu elporti la vidon de tiu malĝojo
kaj vi faris ilin egalaj kun ni, kiuj elportis la ŝarĝon de la tago kaj la varmegon

Vestitor:

Rajzino:zich weten te redden - zich = oneself; weten = be able to; te redden = to save/rescue. So "to be able to save oneself".

It's not about a situation of being able to manage under great duress or difficult circumstances. (...) the connotations of its common English use are in circumstances like how a bereaved person manages feelings after a death or a bout of depression, or bringing up children in a bad situation, i.e when they struggle to have the will or the means.

Yes, I agree that nuance is not captured very well in that translation. I guess I dove so deep into the dictionaries that I forgot to use my gut to feel what the nuances and contexts of words are, because I am familiar with them by intuition.

However, I still think "redden" would be a possibility is some contexts. In a state of desperation a Dutch native might well say "I red het gewoon allemaal niet meer," which is more accurately translated "I just can't handle it all anymore," but I feel that in this context the Dutch "het niet meer redden" conveys more of a struggle to have the will or the means than the English "not being able to handle it anymore". Though admittedly not nearly as much as "Ik kan het gewoon allemaal niet meer aan", which comes a lot closer to expressing these connotations of "cope".

Although the precise usage and connotations of the Dutch words were not as important to the purpose of that post as simply providing another linguistic perspective to look at the word "cope".

Vestitor:

Rajzino:bestrijden - meaning to fight, combat, resist, counteract, control (e.g. a plague). Which I think is not really the nuance we are looking for here.

These second ideas for 'cope with' are not much to do with the meaning of cope as 'managing'. 'Cope with' is not a separate meaning.

I agree. That's why I said it's not what we are looking for. I only mentioned it because I was reporting what my dictionary said.

And yes, the cope in 'cope with' has the same meaning with the preposition as without, but the sentences have another structure when there are no objects to the verb, meaning it sometimes needs different translations in another language. I believe that's what they are there for in the dictionary.
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But this makes me think, the Japanese word 対処, which was the basis for Ceigered's suggestion tajŝi, is also inaccurate.

According to the online Japanese synonym dictionary I frequently use, the more precise meaning of the word is "to deal with an incident with appropriate judgement in accordance to e.g. changes in the situation". The example sentences translate roughly to "to deal/cope with an urgent situation/state of emergency" and "to have made a mistake in the handling of the situation."