It is impossible.
Let the $7\times 28$ area be painted with black and white squares in a checkerboard pattern. Every piece will cover $2$ black and $2$ white squares, except the T-piece, which covers $3$ of one color and $1$ of another. Since there are $7$ T-pieces, a tiling that uses every piece cannot cover the same number of black and white ...
TLDR: I'll fill the board and prove that the solution is unique.
First, let's start by:
I'll paint those green:
Let's repeat those steps a few more times, using orange, blue, red and purple, in precisely that order:
We can easily fill the topmost white squares by that reasoning. They can't be filled in any other way:
Now, let's look at:
And by ...
Since a single line doesn't do damage, it is possible to do
To achieve this, there are a couple of requirements:
To get these pieces one after the other
For this to occur so that the final piece also clears the board, we get these constraints on the number of pieces $X$:
Given these, the smallest $X$ that satisfies both requirements is
This is a small ...
A simple solution involves
Stack the first five arranged the same direction (on the side of the T with the bottom facing either left or right. In my example, the bottom is facing the right). This eliminates the second row (shown with Xs through them). Next, flip the T 180 degrees so the bottom is facing the other direction and fill the gaps as such. This ...
It looks like
This works all nice and fine,
There may, of course, be something obvious I'm missing.
For the J:
With the L, there are some problems that are very similar to the Z case,
Here's the T:
And finally, the S:
And for completeness' sake, the solutions already found by others:
Here is a link to a Mathematics and Computer Science journal published by Kaitlyn M. Tsudura from Saint Mary's University in Halifax, Canada. It summarizes the results of studies published by John Bruztowski and Heidi Burgiel
The studies were basically to determine whether it was possible to make a Tetris game only winnable but they found that it is ...