I have solved the equations with a computer, and I obtained
answered = +-3401222400 (so there are two solutions).
Here are some of the individual values:
z=0 y=+-1 w=1/5 o=y, i=81, e=9,t=10y,n=y/9,x^2=200/964467, etc.
It seems, based on the answers that this problem was one tough nut. Here I offer another approach to it, first, throughout a warm-up example.
We assume that the problem is sensible, that is, there is indeed some solution. Then consider the algebraic expressions A, A, A, and A:
A:= z e r o - 0
A:= o n e - 1
A:= f o u r - 4
A:= t e n - 10
Here these expressions all evaluate to zero, once a solution is substituted in place of the variables.
Consider the following magical weights W, W, W, and W:
W:= f n t u/40
W:= f r t u z (10 - e n t)/400
W:= - z/4
W:= f r u z (o n e t - t - 10*o)/400
Now consider the weighted sum of the four algebraic expressions above:
We look at this expression from two points of view. From one point of view, we substitute in a solution in place of the variables, while from the other point of view we treat this big expression as multi-variate polynomial, unevaluated. So on the one hand, if a solution is indeed exist, then this expression is zero, as all the algebraic expressions A[.] are zero. On the other hand, you can treat this as a BIG algebraic expression in terms of all the involved variables e, f, n, o, r, t, u, z, and direct computation yields (without substituting in anything, that is, without using the fact that A[.] evaluates to zero):
Wow, you must be astonished at this point! This means, that the left hand side is zero, whereas the right hand side is z, that is, we infer from these manipulations that z=0.
Next we address the problem posed by the OP. We assume, that their problem is sensible, that is, there is indeed some solution. Then we consider the algebraic expressions, given by in the OP, as
A := z e r o - 0
A := o n e - 1
A := t w o - 2
A:= m i l l i o n - 1000000
A:= (a n s w e r)^2-p
where A is a new equation, and p:=(a n s w e r)^2 is a new variable. Observe, that if a solution indeed exist, then all of these algebraic expressions evaluate to zero. The task is now to find another expression involving p.
Now imagine, that a wizard gives you some magical weights W, W, ... , W (the omitted expressions have something between 2000 to 20000 terms):
W := 0
W :=-((3099363912*a*d*(29160000 + 7*a*d)*h*n*s*t^3*y)/3125)
W:=11568313814261760 + (347128758144*a*d)/125
And then you consider the weighted sum of the 20 algebraic expressions above, in the exact same way as in the warm-up example, and you consider the BIG expression
from the two viewpoints. On the one hand, this expression must be zero, since all the algebraic expressions A[.] are zero. On the other hand, based on the magic weights, given to you by the wizard, direct computation will testify, that it simplifies to -11568313814261760000 + p. Therefore, you must be astonished once again, since this means that
0 = -11568313814261760000 + p,
and therefore p=(a n s w e r e d)^2=11568313814261760000.
a n s w e r e d = +-3401222400.
Note that this argument does not mean that both signs are possible. However, it shows that the expression a n s w e r e d cannot take any other value.
As for what are the magical weights, and how one should come up with formulae like that in the first place, well... you should ask these questions at our sister site Mathoverflow.SE.