I think that's William Tenn's "Child's Play" (1947). This link has a detailed description of the story, see if this is what you're looking for:
Sam Weber, an unsuccessful lawyer, has received a mysterious box as a Christmas gift - "Merry Christmas, 2353"! Obviously someone is playing pranks.
After much frustration, he will discover the box responds to voice commands - "open" & "close". And its contents are befuddling. "The interior was a crazy mass of shelving on which rested vials filled with blue liquids, jars filled with red solids, transparent tubes... seven pieces of intricate apparatus on the bottom... also a book."
Book is actually a manual. He will learn the box contains something called "Bild-A-Man Set #3... intended solely for the use of children between the ages of eleven & thirteen... will enable the child of this age group to build & assemble complete adult humans in perfect working order". Some of the chapters in the manual:
- "Chapter II - Making simple living things indoors & out".
- "Chapter III - Mannikins and what makes them do the world's work". "Mannikins" are single purpose beings - sort of biological inflexible androids.
- "Chapter IV - Babies & other small humans".
- "Chapter V - Twins for every purpose: twinning yourself & your friends".
- "Chapter VI - What you need to build a man".
- "Chapter VIII - Disassembling the man".
The manufacturer is "The Bild-A-Man Company, 928, Diagonal Level, Glunt City, Ohio." Of course, inquiries will reveal no such firm or address exists.
And you can read the whole thing on Scribd.
That's the same plot, and the man also makes a couple of monsters, a baby who's an almost perfect replica of the one he's babysitting, and is planning to clone a woman he's interested in using the Bild-a-man device - does this sound right?