The points that satisfy the system $ x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25,$ constitute the following set:

$ \textbf{(A)}\ \text{only two points} \qquad \\
\textbf{(B)}\ \text{an arc of a circle}\qquad \\
\textbf{(C)}\ \text{a straight line segment not including the end\minus{}points}\qquad \\
\textbf{(D)}\ \text{a straight line segment including the end\minus{}points}\qquad \\
\textbf{(E)}\ \text{a single point}$
1. Consider the system of equations:
   \[
   x + y = 1
   \]
   \[
   x^2 + y^2 < 25
   \]

2. The first equation, \(x + y = 1\), represents a straight line in the coordinate plane. This line can be rewritten as:
   \[
   y = 1 - x
   \]

3. The second inequality, \(x^2 + y^2 < 25\), represents the interior of a circle with radius 5 centered at the origin \((0,0)\). The boundary of this circle is given by:
   \[
   x^2 + y^2 = 25
   \]

4. To find the points that satisfy both conditions, we substitute \(y = 1 - x\) into the circle's equation:
   \[
   x^2 + (1 - x)^2 < 25
   \]
   Simplifying the left-hand side:
   \[
   x^2 + (1 - 2x + x^2) < 25
   \]
   \[
   x^2 + 1 - 2x + x^2 < 25
   \]
   \[
   2x^2 - 2x + 1 < 25
   \]
   \[
   2x^2 - 2x + 1 - 25 < 0
   \]
   \[
   2x^2 - 2x - 24 < 0
   \]
   \[
   x^2 - x - 12 < 0
   \]

5. To solve the quadratic inequality \(x^2 - x - 12 < 0\), we first find the roots of the corresponding equation \(x^2 - x - 12 = 0\):
   \[
   x^2 - x - 12 = 0
   \]
   Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -1\), and \(c = -12\):
   \[
   x = \frac{1 \pm \sqrt{1 + 48}}{2}
   \]
   \[
   x = \frac{1 \pm \sqrt{49}}{2}
   \]
   \[
   x = \frac{1 \pm 7}{2}
   \]
   \[
   x = 4 \quad \text{or} \quad x = -3
   \]

6. The quadratic inequality \(x^2 - x - 12 < 0\) is satisfied for \(x\) values between the roots:
   \[
   -3 < x < 4
   \]

7. Since \(y = 1 - x\), the corresponding \(y\) values are:
   \[
   y = 1 - x \quad \text{for} \quad -3 < x < 4
   \]
   This gives:
   \[
   1 - 4 < y < 1 + 3
   \]
   \[
   -3 < y < 4
   \]

8. Therefore, the points \((x, y)\) that satisfy both conditions lie on the line segment \(x + y = 1\) but within the circle \(x^2 + y^2 < 25\), excluding the endpoints where \(x^2 + y^2 = 25\).

Conclusion:
The points that satisfy the system constitute a straight line segment not including the endpoints.

The final answer is \( \boxed{ \textbf{(C)} } \) a straight line segment not including the endpoints.