Why is there no PRODUCT aggregate function in SQL?

The problem can be solved using modern SQL features such as window functions and CTEs. Everything is standard SQL and - unlike logarithm-based solutions - does not require switching from integer world to floating point world nor handling nonpositive numbers. Just number rows and evaluate product in recursive query until no row remain:

   with recursive t(c) as (
     select unnest(array[2,5,7,8])
   ), r(c,n) as (
     select t.c, row_number() over () from t
   ), p(c,n) as (
     select c, n from r where n = 1
     union all
     select r.c * p.c, r.n from p join r on p.n + 1 = r.n
   )
   select c from p where n = (select max(n) from p);

As your question involves grouping by sale column, things got little bit complicated but it's still solvable:

   with recursive t(sale,price) as (
     select 'multiplication', 2 union
     select 'multiplication', 5 union
     select 'multiplication', 7 union
     select 'multiplication', 8 union
     select 'trivial', 1 union
     select 'trivial', 8 union
     select 'negatives work', -2 union
     select 'negatives work', -3 union
     select 'negatives work', -5 union
     select 'look ma, zero works too!', 1 union
     select 'look ma, zero works too!', 0 union
     select 'look ma, zero works too!', 2
   ), r(sale,price,n,maxn) as (
     select t.sale, t.price, row_number() over (partition by sale), count(1) over (partition by sale)
     from t
   ), p(sale,price,n,maxn) as (
     select sale, price, n, maxn
     from r where n = 1
     union all
     select p.sale, r.price * p.price, r.n, r.maxn
     from p
     join r on p.sale = r.sale and p.n + 1 = r.n
   )
   select sale, price
   from p
   where n = maxn
   order by sale;

Result:

sale,price
"look ma, zero works too!",0
multiplication,560
negatives work,-30
trivial,8

Tested on Postgres.

Another approach based on fact that the cardinality of cartesian product is product of cardinalities of particular sets ;-)

⚠ WARNING: This example is just for fun and is rather academic, don't use it in production! (apart from the fact it's just for positive and practically small integers)⚠

with recursive t(c) as (
  select unnest(array[2,5,7,8])
), p(a) as (
  select array_agg(c) from t
  union all
  select p.a[2:]
  from p
  cross join generate_series(1, p.a[1])
)
select count(*) from p where cardinality(a) = 0;

Here is an oracle solution for anyone who needs it

with data(id, val) as(
select 1,1.0 from dual union all
select 2,-2.0 from dual union all
select 3,1.0 from dual union all
select 4,2.0 from dual 
),
neg(val , modifier) as(
select exp(sum(ln(abs(val)))), case when mod(count(*),2) = 0 then 1 Else -1 end
from data
where val <0
)
,
pos(val) as (
select exp(sum(ln(val)))
from data
where val >=0
)
select (select val*modifier from neg)*(select val from pos) product from dual