One alternative is to plot several points along the circle using Cartesian coordinates. This has already been suggested, but it can be generalized to deal with many kinds of obstacles.
If you did have free access to all points in the interior of the circle, you could set up your Cartesian coordinate system by laying two straight reference lines perpendicular to each other to form two diameters of the circle, like so:
For a circle of radius $r$, you could make a table of $x$ and $y$ values using the formulas $x = r \cos \theta$ and $y = r \sin \theta$ for a sequence of angles.
For each pair $(x,y)$, you measure a distance $x$ along one of your reference lines to find point $A$ and a distance $y$ along the other reference line to find point $B$.
Then attach a string of length $x$ at $B$ and a string of length $y$ at $A$ and extend the two strings taut so that they meet at $C$, which is a point on the circle.
After plotting several points regularly spaced around the circle, you can use a curved template (constructed elsewhere) to mark the arc of the circle between each pair of points. The number of points you need to plot is a function of how long you can make your template relative to the radius of the circle. For example, if you can build a template that is slightly more than a $10$-degree arc of the circle, you only have to use a table where the angle $\theta$ is given in $10$-degree increments.
Now to generalize: the two reference lines do not need to be diameters. You can offset one or both of the lines from the center of the circle, and as long as they are perpendicular to each other, you merely need to add or subtract the amount of each offset from the relevant coordinate. For example, let's move one of the reference lines $a$ feet from the center and the other reference line $b$ feet from the center. The result looks like this:
Now to plot the point that was at $(x,y)$ in your original table of coordinates, you place points $A$ and $B$ at distances $|x-a|$ (or $x+a$) and $|y-b|$ (or $y+b$) along the reference lines, measured from where the lines cross,
and then measure the same distances from the points $A$ and $B$ in order to find
a point $C$ on the circle.
Now, for example, to deal with the shed in the middle of the circle, you lay your reference lines along two sides of the shed. The distances $a$ and $b$ in this case are just half the dimensions of the shed. You can then plot more than half the circumference of the circle. To plot the remaining part of the circle, lay reference lines along the other two sides of the shed.
For the circle cut off by the neighbor's fence, lay one reference line along the fence (or use the fence itself if it is straight enough) and lay the other reference line perpendicular to the first. In this case you only need one of the lines (the one parallel to the fence) to be offset from the center, which simplifies the task; you can plot two symmetric points using each $(x,y)$ pair.
For the patio at the edge of the pool you might find it convenient to lay the two perpendicular reference lines so that they are tangent to the circle you want to plot. That is, the offsets are $a = r$ and $b = r$.
For the planters, you could (in the example shown) lay the two reference lines just a bit off-center while staying clear of all the planters, but then you might not be able to plot as many points as you might like (because planters would interfere with the lines from $A$ or $B$ to $C$). You might find it easier to lay out a square larger than the circle, but with the same center, and use adjacent sides of that square as reference lines for the various parts of the circle. That is, you can let $a = b > r$.
But in the example shown, there is one arc of the circle that cannot be plotted by any pair of sides of the square, so you would have to lay another reference line between the planters, perpendicular to the nearby side of the square, in order to plot that arc.
By the way, for a circular curved driveway of uniform width, if you make templates for both the inner and outer radii of the driveway and attach them to a rigid frame so that their arcs are concentric (perhaps by attaching plywood "blanks" to the frame, drawing the arcs on them from a common center point, and then cutting the arcs), you need only plot points along one edge of the driveway; every time you use the template to fill in the points along that arc, you can use the other side of the template to fill in points along the other arc of the driveway.