Your case is an example where not the diagonal elements, but the distances hiding behind the coefficients are to blame for the not-positive-definiteness of the matrix. I described it here as "Cause 1" (displayed on Fig. 1). For short, your 7 items as points in space do not converge in Euclidean space. That cause is unlikely to be cured by adding a constant to the diagonal of the covariance matrix. You said it did not help you. Without checking your words, I tried to add a constant to all the elements of the matrix; it didn't help either.
Below are the outputs of what I did with your data then. First shown are the eigenvalues of your matrix as it is. I converted the matrix into Euclidean distance matrix by means of the Law of cosines.
I then performed double centering of that distance matrix. Double centering finds geometric centroid of the cloud of points between which the distances are, and converts the distances into new covariate-like scalar products. The resultant matrix describes the distances, but their entries are angle-type similarities, so we are in right to eigen-decompose it the usual way. I don't show the doubly-centered matrix itself, but I show its eigenvalues. Note that there are negative ones, their presence is the evidence of "Cause 1" of not-positive-definiteness.
So, the distances, that are behind the covarinces, should be inflated with a small constant. I added the constant, and the thus corrected Euclidean distances, when doubly-centered, showed no negative eigenvalues. Good.
Now, what we need is to transform distanced back to covariances. We do it again by the Law of cosines, using the diagonal of the input covariance matrix. The resultant covariance matrix has this diagonal and has corrected off-diagonal values, and is now positive definite.
Eigenvalues of the input cov matrix:
139.9695727
.0114858
.0000673
.0000601
-.0000215
-.0000530
-.0001113
Euclidean distances:
.00000 .02000 .04000 .06083 .08124 .10100 .12166
.02000 .00000 .02000 .04123 .06000 .08124 .10100
.04000 .02000 .00000 .01732 .04000 .06000 .08124
.06083 .04123 .01732 .00000 .01732 .04123 .06083
.08124 .06000 .04000 .01732 .00000 .02000 .04000
.10100 .08124 .06000 .04123 .02000 .00000 .02000
.12166 .10100 .08124 .06083 .04000 .02000 .00000
Eigenvalues of the double centered matrix of distances:
10 ** -2 X
1.148575892
.006728980
.006090536
.000000000
-.002108362
-.005304872
-.011125031
Corrected Euclidean distances:
.00000 .02965 .04965 .07048 .09089 .11065 .13131
.02965 .00000 .02965 .05088 .06965 .09089 .11065
.04965 .02965 .00000 .02697 .04965 .06965 .09089
.07048 .05088 .02697 .00000 .02697 .05088 .07048
.09089 .06965 .04965 .02697 .00000 .02965 .04965
.11065 .09089 .06965 .05088 .02965 .00000 .02965
.13131 .11065 .09089 .07048 .04965 .02965 .00000
Eigenvalues of the double centered corrected matrix of distances:
.013491933
.000636840
.000269820
.000222552
.000191635
.000000993
.000000000
Corrected cov matrix:
19.99390 19.99516 19.99537 19.99447 19.99247 19.98948 19.98528
19.99516 19.99730 19.99786 19.99736 19.99587 19.99317 19.98948
19.99537 19.99786 19.99930 19.99929 19.99807 19.99587 19.99247
19.99447 19.99736 19.99929 20.00000 19.99929 19.99736 19.99447
19.99247 19.99587 19.99807 19.99929 19.99930 19.99786 19.99537
19.98948 19.99317 19.99587 19.99736 19.99786 19.99730 19.99516
19.98528 19.98948 19.99247 19.99447 19.99537 19.99516 19.99390
Its eigenvalues:
139.9661876
.0134919
.0006356
.0002698
.0002226
.0001915
.0000010