Well, the same is true for any irrational number, and there are an infinite number of irrational numbers between any two rational numbers. The exceptional thing about our DNA, though, is not that it contains information, but that it contains systematically organized information structures. The decimal form of an irrational number, though, doesn’t provide much if any useable information, much less anything like the purposeful structuring of a DNA code. The infinite decimals are simply an artifact of performing a mathematical operation in a number system unable to render the result with precision. If we use fractions, they all vanish into thin air :)
It is actually an open problem whether every finite sequence of digits occurs in the decimal expansion of pi (and likewise open for expansions in any other base).
But it is most assuredly not the case that all irrational numbers have every sequence of digits, for instance the number with decimal expansion
there being n 4’s between the n-th 5 and the (n+1)-st 5 is irrational, but only contains a very restricted set of finite digit sequences — in particular having only 4’s and 5’s as digits, but even more restricted than that: any sequence with a single 5 and any number of 4’s before and after will occur, but once two 5’s occur (necessarily non-adjacent) the first can be preceded by at most one fewer 4’s than occur between the left-most 5 and the next 5, the number of 4’s between successive 5’s must increase by one and the number of 4’s after the right-most 5 must be at most one more than the number of 4’s between the two right-most 5’s.
Actually, like my example, pi is a very tame irrational number: though less obviously so than for my example, its decimal expansion can, in principle, be generated by running an algorithm. Like the rational numbers, the set of real numbers, both rational and irrational, whose decimal (or binary or base b) expansion can be generated by an algorithm is countable, so there are uncountably many “non-algorithmic” irrationals (my favorite being the “oracular number for FORTRAN”, defined by listing all valid FORTRAN programs ordered by total number of symbols and within each number in lexicographic order by the standard ordering on ASCII symbols, the i-th digit right of the binary point is 1 if the i-th program would halt in finite time when run on an machine with infinite memory, and 0 otherwise). That, and its friends, other binary, decimal and base b oracular numbers for various and sundry ways of describing algorithms, are some of the very few non-algorithmic irrationals one can describe at all.
I assume you are alluding to CONTINUED fractions? Many irrationals have a simple repetitive continued fraction representation, and even 'e' has a regular pattern, even though it doesn't actually repeat. And of course any finite continued fraction is a simple fraction, and can be represented by a repeating decimal.