Your photo has low resolution. Its good for onscreen purposes, but it allows high quality printing only as about 2,5 cm wide total. Fortunately we can easily increase it to 7,5 cm without very bad quality loss.
The ellipses seem to be even more unsharp than the low resolution alone would cause. We can fix the ellipses at the same time when we remove the backgroud. I assume that you want to save te white inside the ellipses as the color of the smaller shapes.
1.Take the elliptical selection tool and draw a selection over the leftmost ellipse. Goto Select > Transform Selection and drag the ellipse perfect. finally press enter to fix the selection for use.
Take the smudge tool. Push the blue color against the selection around the selected area. You fix a little of the unsharpness.
Press Ctrl+C (=copy), goto Edit > Paste Special > Paste in Place. You get the left part as a new layer without the background (in the screensot the selection marquee makes the border unclear)
- Do the same with the right part. You have now the logo in 2 layers without the background. The original layer is disabled in the following screenshot.
- Merge the left and right part layers, delete the original layer and crop the image to be not much bigger than the logo, Now it's ready to save as PNG.
6 We have one trick left: Enlargening. It needs appropriate software, in this case ON1 perfect resize. It can enlargen drawings and often even photos without the same loss of apparent sharpness than scaling the image bigger in Photoshop. Of course it quesses the missing detail, but in easy cases it succeeds. Here's an example. The enlargening (300%) is done first. The ellipse borders are cleaned by pushing the color with smudge against the selection when the halves were copied. Not perfect, but maybe usable:
Warning: This is PNG with transparent background. If you copy it and paste it to Photoshop, you probably get black background. Download it, then it works better.