I get 355 / 113 using the following method:

SET NOCOUNT ON
DECLARE @numbers TABLE (ID NUMERIC(18, 14) PRIMARY KEY)

DECLARE @i INT
SET @i = 1
  
WHILE @i < 1000
BEGIN
    INSERT INTO @numbers
    VALUES(@i)
    SET @i = @i + 1
END

;WITH CTE
AS
(
SELECT  ROW_NUMBER() OVER (ORDER BY ABS(PI() - n1.ID / n2.ID)) AS RowNumber,
        n1.ID / n2.ID AS Ratio,
        CAST(n1.ID AS INT) AS N1ID,
        CAST(n2.ID AS INT) AS N2ID
FROM @numbers n1
    CROSS JOIN @numbers n2
)
SELECT  *,
        PI() AS PI
FROM CTE
WHERE RowNumber = 1

...and here's a slightly tweaked version that will return a pair of values that cannot be reduced:

As I understand Alex's puzzle. you have to actually calculate PI. If you already know PI, then it is easy to calculate the best-fit numerator and denominator below a particular limit.

DECLARE @numbers TABLE
   (n INT NOT NULL IDENTITY PRIMARY KEY)
INSERT @numbers DEFAULT
        VALUES ;
WHILE SCOPE_IDENTITY() < 1000
    INSERT @numbers DEFAULT
        VALUES ;
SELECT  TOP 1
         n AS numerator,
         ROUND(n/PI(),0) AS denominator,
         n/ROUND(n/PI(),0) AS Decimal_Fraction  
    FROM @numbers
    WHERE n>1
    ORDER BY ABS(PI()-n/ ROUND(n/PI(),0)) ASC,
             n ASC
            

DECLARE @numbers TABLE
  (n INT NOT NULL IDENTITY PRIMARY KEY,
   approximation numeric(19,18) DEFAULT 0)
INSERT @numbers DEFAULT VALUES ;
WHILE SCOPE_IDENTITY() < 1000
  INSERT @numbers DEFAULT VALUES ;

DECLARE @PI numeric(19,18),
  @Approximation numeric(19,18),
  @operator CHAR(1)
SELECT  @operator='+', @approximation=0

UPDATE @numbers
  SET @Approximation=approximation=@Approximation +
  CASE WHEN @operator='+' THEN +(4.00/n)
                    ELSE -(4.00/n) END,
  @operator = CASE WHEN @operator='+' THEN '-'
                    ELSE '+' END
  WHERE n % 2=1
SELECT @PI=AVG(approximation)
  FROM @numbers WHERE n % 2=1 AND n>991

SELECT  TOP 1
      n AS numerator,
      ROUND(n/@PI,0) AS denominator,
      n/ROUND(n/@PI,0) AS Decimal_Fraction
  FROM @numbers  WHERE n>1
  ORDER BY ABS(@PI-n/ ROUND(n/@PI,0)) ASC,
      n ASC
      

>> Consider the following brainteaser: using only commands, features and functions described in Books Online except for PI(), find the best approximation of Pi number as a ratio of two integer numbers, both less than 1000. The ideal solution is a ratio that cannot be reduced. You can use a helper table populated with integer numbers. <<

A little research tells us that 355/113 is good to six decimals and meets the criteria. But 103993/33102 is good to nine decimals, so we can get a better approximation of pi without any fancy math at all.

The fastest way is to just use constants in a simple query:

I did it as decimal and not float to keep this at the level of a pocket calculator. If I can do more math than just a simple ratio or use floats, there are better formulas.

I suppose we could do a brute force query like:

Digits builds the table of 1 to 1000 integers. PiList is a table of candidate approximations. The base query is the (dividend, divisor) pair closet to the better aproximation. We get (355, 113) and (710, 226) as candidates, so we could do one more CTE of Candidates and add a (ROW_NUMBER() OVER (dividend)) AS candidate_size to it and keep only (candidate_size = 1).