I have a suggestion: reset your time origin. It looks like your t axis is in years (1981, 1982, etc.). So, instead of plugging in 1988, which you must then raise to the 8th power, try a time axis shift: tau = t - 1980. Now, you'll be plugging in the small tau's to be exponentiated, instead of the large years. That might help condition the matrix a bit. For example, using the code

f[x_] := Aye x^8 + Bee x^7 + Cee x^6 + Dee x^5 + Eee x^4 + Eff x^3 + Gee x^2 + H x + Eye;

Solve[{f[0] == 426.8, f[1] == 470.2, f[2] == 503.4, f[3] == 557.3, f[4] == 564.7, f[5] == 575.4, f[6] == 579.8, f[7] == 624.7, f[8] == 669.9}, {Aye, Bee, Cee, Dee, Eee, Eff, Gee, H, Eye}]

Mathematica spit out an answer that did not suffer from being ill-conditioned. Now, you can re-shift your answer back to find the correct numbers with the years.