Solution:

Exponentiating each equation gives $xyz - 3 + \log_5 x = 2^5$ and similarly for the others, so $$xyz + \log_5 x = 35, \qquad xyz + \log_5 y = 84, \qquad xyz + \log_5 z = 259.$$ Write $x = 5^a,$ $y = 5^b,$ $z = 5^c,$ so that $xyz = 5^{a+b+c}.$

Adding the three equations gives $3 \cdot 5^s + s = 378,$ where $s = a + b + c.$ The left side is strictly increasing in $s,$ and $s = 3$ works since $375 + 3 = 378,$ so $s = 3$ and $5^s = 125.$

Then $a = 35 - 125 = -90,$ $b = 84 - 125 = -41,$ and $c = 259 - 125 = 134,$ so $|a| + |b| + |c| = 90 + 41 + 134 = 265.$