"That both creates and embraces who we are" could refer to "a sense of balance", but there is no grammatical reason why it should not refer to "that rich inner and interpersonal life". Personally I think it's more likely to be the second, particularly if it's in line with what the author says elsewhere.

As Yvonne has explained, you could read it as "a sense of balance that both creates and embraces who we are", and take "within that inner and interpersonal life" as qualifying "a sense of balance". Alternatively, you could perfectly well take "that rich inner and interpersonal life that both creates and embraces who we are and who we can become" as a single unit. Neither reading can be excluded and both make grammatical sense. It's just a question of which is more likely. I think it's more likely — much more likely, in fact — that what creates and embraces our identity is our inner and interpersonal life itself rather than a sense of balance within that inner and interpersonal life. A sense of balance is no doubt desirable, but I don't think that sense of balance creates our identity; I think it's our life itself.

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Note added at 19 hrs (2018-05-23 09:59:15 GMT)
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I am quite sure that there is no grammatical reason why the second "that" can't refer to "life". The main clause, reduced to essentials, is:

"Equanimity is the ability to [...] attain a sense of balance within that [...] life."

Then "that both creates and embraces who we are and who we can become" is a relative clause qualifying "that [...] life". Why not?

Suppose we rephrase the same sentence with "activities" instead of "life":

"And equanimity is the ability to [...] attain a sense of balance within those rich inner and interpersonal activities that both create and embrace who we are and who we can become."

Here, obviously, "that create and embrace" can't refer to "balance", because the verbs are plural. They must refer to "activities". But the sentence is perfectly grammatical.

Alternatively, suppose it were like this:

"And equanimity is the ability to [...] attain a sense of balance within those rich inner and interpersonal activities that both creates and embraces who we are and who we can become."

This, too, is perfectly grammatical, but the verbs are singular, so the relative clause must refer to "balance".

Given that both are syntactically possible, the question, as I say, is which is more likely to be what the author meant: which is more likely to be what creates and embraces who we are: our inner and interpersonal life or a sense of balance within that inner and interpersonal life? I say the former.

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Note added at 19 hrs (2018-05-23 10:16:21 GMT)
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"That rich inner and interpersonal life that both creates and embraces who we are and who we can become" can perfectly well be a single unit, a noun phrase. We could say, for example:

"We need to focus all our attention on that rich inner and interpersonal life that both creates and embraces who we are and who we can become."

The relative clause can only refer to "life" here. But there's nothing grammatically wrong with the sentence.

Or you could even make it the subject:

"That rich inner and interpersonal life that both creates and embraces who we are and who we can become requires us to attain a sense of balance."

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Note added at 21 hrs (2018-05-23 12:40:14 GMT)
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I don't think Yvonne is using American English; she's from Ireland, and I don't think there are any differences between British and Irish English grammar in this case. I can't really comment further because I still don't understand why she is so sure.

There is a slight difference between American and British English in the use of relative pronouns, specifically in the use of "which", though I don't think it's an issue here:

"Is there any difference between the use of that and which in sentences such as any book that gets children reading is worth having, and any book which gets children reading is worth having? The general rule in British English is that, in restrictive relative clauses, where the relative clause serves to define or restrict the reference to the particular one described, which can replace that. However, in non-restrictive relative clauses, where the relative clause serves only to give additional information, that cannot be used: this book, which is set in the last century, is very popular with teenagers but not this book, that is set in the last century, is very popular with teenagers. In US English which is generally used only for non-restrictive relative clauses."
https://en.oxforddictionaries.com/definition/that#that__61

I think that it's debatable whether this relative clause is restrictive or non-restrictive, and equally debatable whether it refers to "balance" or to "life". To my ear, "which" could have been used instead of "that", since "that both creates and embraces who we are and who we can become" could be regarded as providing original information about "balance" or "life" rather than defining or restricting the reference. But as I say, this factor applies equally to both readings, in my opinion.